The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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How can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\mathrm{End}V)[[x,x^{-1}]]$?

In the context of vertex operator algebras, if $V$ is a vector space, how can we view $\operatorname{Hom}(V,V((x)))$ as a subspace of $(\operatorname{End}V)[[x,x^{-1}]]$? The notation $V((x))$ is the ...
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23 views

Two identities concerning reduced crossed products

I am currently writing a project wherein I need two identifications. Let's fix some notation. Let $\Gamma$ be a discrete group acting on a unital C*-algebra A by $\ast$-automorphisms $\alpha: A ...
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52 views

Hermitian Pairings from Positive Functionals

Let $A$ be $*$-algebra and $\phi:A \to {\mathbb C}$ a positive linear functional, that is, one for which $\phi(aa^*) \geq 0$, for all $a \in A$. When does it hold that a symmetric sesquilinear form, ...
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1answer
32 views

Positive Map: Reduction

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. (Both possibly nonunital!) Linear Map: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\varphi\in\mathcal{L}$$ Implication: ...
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79 views

Most natural equivalence between $C^*$-algebras

I have listen or read that, in the context of noncommutative geometry, Morita equivalence is a more natural equivalence for $C^*$-algebras than $*$-isomorphism. Can someone explain this sentence or ...
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19 views

What are some specific operations for matrices, like is dot product for vectors?

I read this question and the associated answer (What is the dot product between a vector of matrices?), and it wasn't what I am looking for. I want to know if there are some special operations ...
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16 views

about a definition in the representation of $L^1(G)$, where $G$ is locally compact

I try to learn group C*-algebra by reading the Chapter 7 in the book C*-algebras by example. It says that if $\pi$ is a unitary representation of $G$, then it induces a representation of $L^1(G)$ by ...
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24 views

When set of projections is closed under multiplication

Let $\mathcal A\subset B(H)$ be an unital $C^*$ algebra of operators on a Hilbert space $H$. Let's denote by $\mathcal P$ the set of projections in $\mathcal A$, that is $\mathcal P:=\{a\in \mathcal A ...
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24 views

Relation of fundamental group $\pi_1(X,x_0)$ and properties of C(X)

As stated by commutative Gelfand Naimark theorem, every unital C* algebra is of the form C(X) for some compact Hausdorff space X. Moreover two such algebras are isomorphic iff the corresponding ...
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22 views

irreducible implies the commutant consists of multiples of identity?

I was trying to solve exercises (4) on Page 59 of the book "A short course on spectral theory", William Avreson. Let $A$ be a Banach star-algebra. A representation $\pi\in$rep$(A,H)$ is said to be ...
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1answer
39 views

Uniqueness of positive square root of postive element in C* algebra

If a is a positive element then it has a unique positive square root, i.e. a unique b positive such that b^2=a. I understand the existence part of the proof. If ...
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71 views

Classification of vector bundles over the torus

In M. Rieffel's paper "The Cancellation Theorem for projective modules over irrational rotation $C^*$-algebras", he classifies finitely generated projective modules over the $C^*$-algebra ...
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What is this operator topology?

Let $X$ be a separable Banach space with (norm $1$) Schauder-Basis $\{e_n\}_{n\in\mathbb N}$. Denote for $x\in X$ with $|\cdot|_x$ the seminorm on $\mathcal L(X)$ given by $|A|_x = \|A x\|$. Consider: ...
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89 views

Proving an isometric dilation of a non unitary operator on Hilbert space implies infinite dimensional space involving matrices

I have been given this exercise in my Operator theory class dealing with operators on Hilbert spaces, which reads as follows: Let H be a Hilbert space. We are to prove, in two distinct ways, that ...
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41 views

Singular value decomposition of sum of single particle operators

here is my question. Suppose to have an operator $L$ in a composite Hilbert space $A\otimes B$ which can be written as sum of single particle operator as: \begin{equation} L = (L_0\otimes \mathbb{I} + ...
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1answer
30 views

Are Baer Rings worth studying for significant modern progress in vNa's?

I am wondering if Kaplansky's "Ring of Operators" is worth studying if I'm interested in functional analysis (more specifically von Neumann algebras). Yes any vNa is a B. ring but my question is more ...
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1answer
28 views

Explicit example of Gel'fand transform

I would like to determine explicitly the Gel'fand transform for the (commutative unital) Banach algebra of $2 \times 2$ matrices of the form $$\pmatrix{ a && b \\ 0 && a}, \quad a,b ...
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$C^\ast$ condition implies $B^\ast$ condition

By $C^\ast$ condition I understand $\|A^\ast A\|=\|A^\ast\|\|A\|$ and for $B^\ast$, $\|A^\ast A\|=\|A\|^2$. I know these conditions are equivalent even NOT assuming the involution is isometric, but I ...
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14 views

Do a von Neumann Algebra's subalgebras have unique “complements”?

Let $A \subset B \subset C$ be von Neumann algebras (and more specifically factors, if it helps) either all type II or all type III, acting on the same separable Hilbert space. Call $A' \subset B$ ...
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2answers
63 views

Morphism: Unitization

Given C*-Algebras $\mathcal{A}$ and $\mathcal{B}$. (Possibly unital!) Morphisms are contractive: $$\varphi:\mathcal{A}\to\mathcal{B}:\quad\|\varphi\|\leq1$$ (Possibly nonunital!) How to apply ...
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44 views

A point-wise separation Hahn-Banach theorem in C*-algebras

Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$. Let $E$ be a norm closed convex subset of positive operators in $K(H)$ ...
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Is this a characterization of commutative $C^{*}$-algebras

Assume that $A$ is a $C^{*}$-algebra such that $\forall a,b \in A, ab=0 \iff ba=0$. Is $A$ necessarily a commutative algebra? In particular does "$\forall a,b \in A, ab=0 \iff ba=0$" ...
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1answer
28 views

Where does Gelfand Theory fail for non-commutative algebras.

I'm trying to get my head around Gelfand theory, and I can't seem to find the subtleties between commutative and non-commutative algebras. Why is there not a one-to-one correspondence between maximal ...
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36 views

Commuting Matrices and Operatos

I'm a little confused as to how to commute (specifically that induced by the transpose in Hermitian conjugation) matrices when one contains operators and is pre-multiplied with the other. For ...
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A nilpotent element of an algebra which does not lie in the span of commutator elements.

What is an example of a $C^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements. Obviously any such $C^{*}$ algebra would be a ...
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1answer
21 views

Does pointwise nilpotency imply global nilpotency?

Is there a compact Haussdorf space $X$ and $C^{*}$ algebra $A$ with a continuous map $f:X\to A$, such that $f(x)\in A$ is a nilpotent element, $\forall x \in X$, but $f$ is not a ...
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24 views

Equivalence of Definitions of Strong Operator Topology

I have a couple questions about how we define the strong operator topology on $\mathscr{B} (H)$ that I'm hoping someone can help me with. First, I thought that the strong operator topology was the ...
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1answer
44 views

Homotopy elements in C*-algebras

Let A is a C*-algebra, A$^+$ means A$\times$$\mathbb{C}$ equipped with pointwise sum and with a multiplication defined by: (a, $\lambda$)(b, $\mu$) = (ab + $\lambda$b + $\mu$a, $\lambda$$\mu$) If x, ...
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35 views

Atomic W*- algebra

Let $A$ be a C*-algebra. Put $z: = \sup\{ e\in A^{**} ;\text{ e is a minimal projection}\}$. Easily can see $z$ is a central projection. Set $M:= A^{**}z$. 1) Is $M$ an atomic W*-algebra in general? ...
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65 views

$C(S^1)$ does not have a single generator

Let $S^1$ be the unit circle in the complex plain and $C(S^1)$ be the continuous function space on $S^1$.$f\in C(S^1)$ is a generator means that $\{p(f) |\text{ p is a polynomial in z}\}$ is dense in ...
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Direct limits of simple C*-algebras are simple

Let $S$ be a non-empty set of simple C$^*$-subbalgebras of a C$^*$-algebra $A$. Let us also suppose that $S$ is upwards-directed and that the union of all element of $S$ is dense in $A$. Then $A$ is ...
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1answer
37 views

How is the spectrum of a product of operators related to the spectrum of each term in the product?

I will use the usual notation of $\sigma(A)$ to denote the spectrum of an operator $A$. Is there a relationship between the spectrum of bounded operators (on complex Hilbert space) $A$ and $B$ and ...
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91 views

Comparing two positive linear functionals when support$(\psi)\leq$ support$(\phi)$

The following point has been addressed in the significant paper "order ideal in C*-algebras and its dual (By E. Effros - Lemma 4.1). The reference is Dixmier's book (1957 French). Unfortunately, I ...
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2answers
48 views

Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\# $ is ...
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1answer
53 views

$C*$-algebra Identity for Matrices

What's the easiest way to see that the $n \times n$-matrices over $\mathbb{C}$ satisfy the $C^*$-algebra identity $\|aa^*\| = \|a\|^2$?
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One-sided identities in Banach algebras

What is an example of a Banach algebra with a left identity but with no right identities? Is there an example of an operator algebra with this property?
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Special case of Green's imprimitivity theorem and related question

Consider a locally compact group $G$ and a closed subgroup $H$ of $G$, and let $G$ act on $G/H$ by left translation. Green's imprimitivity theorem implies that the crossed product $C_0(G/H)\rtimes G$ ...
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65 views

The reduction of nilpotency order of nilpotent elements of $C^{*}$ algebras

Assume that $A$ is a unital $C^{*}$-algebra. Let $a\in A$ be a nilpotent element with $$a^{k}=0,\;\;k>1.$$ Are there two elements $x,y\in A$ with $a=xy,\;\;(yx)^{k-1}=0$? Motivation for ...
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1answer
29 views

Expectation value of the product of two operators

Does $\langle A^2B^2\rangle = \langle B^2A^2\rangle$ (i.e. the operators $A$ and $B$ commute) somehow imply $\langle A^2\rangle = \langle B^2\rangle$? If so, why? To add some context, my operators ...
2
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1answer
40 views

A certain *-isomorphism

let $A$ be a C*-algebra and $z\in A^{**}$ the supremum of all the minimal projections in $A^{**}$. How can show $* -$ homomorphism $A\to zA\subset zA^{**}$ is injective?
3
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Understanding Operators in context of Green's function derivation

I am trying to understand what operators actually mean when deriving the definition of green's function. What is the interagl representation of an operator? Is this correct? $$ D = <x|\int D|x> ...
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1answer
74 views

Is the algebra of adjointable operators on a Hilbert module prime?

In abstract algebra, a nonzero ring $R$ is a prime ring if for any two elements $a$ and $b$ of R, $arb = 0$ for all $r$ in $R$ implies that either $a = 0$ or $b = 0.$ Or for any two ideals $A$ and ...
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Is there a sample-path continuous stochastic process whose sample paths do not almost surely lie in an RKHS?

Let $f$ be a mean zero second-order stochastic process with continuous covariance function $k$, that is indexed on a separable metric space $\mathcal{X}$ and that is sample-path continuous. Can we ...
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1answer
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Can $xy$ and $yx$ lie in different connected components of the group of invertible elements of an algebra?

What is an example of a Banach or $C^{*}$ algebra $A$ which has two invertible elements $x, y$ such that $xy$ can not be connected to $yx$ in $G(A)$, the space of invertible elements of $A$. A ...
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1answer
71 views

Can all compact subsets of $\mathbb{C}$ be spectra of a bounded operator on $C[0,1]$?

Let $K$ be a non-empty compact subset of $\mathbb{C}$, the complex field. Does there exist an operator $A\in \mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is ...
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21 views

Binormal operator - equivalent definitions?

I have seen two different definitions of a binormal operator A. A is unitarily equivalent to a block 2x2 matrix of commuting normal matrices. AA* commutes with A*A. I am hoping these definitions ...
2
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2answers
106 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
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1answer
59 views

The algebras of compact operators on $\ell_p$ as direct limits of matrix algebras

Consider $M_n(\mathbb{C})$ as $B(\ell_p^n)$ for $n\in\mathbb{N}$ where $p\in[1,\infty)$, and include $M_n(\mathbb{C})$ in $M_{n+1}(\mathbb{C})$ as the upper left corner. Is it true that ...
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1answer
68 views

Does $C^*(G) \cong C^*(H)$ imply that $\mathbb{C}G \cong \mathbb{C}H$?

I wonder whether the underlying complex group algebra of a group $C^*$-algebra is unique? I.e. if $G$ and $H$ are discrete groups such that $C^*(G) \cong C^*(H)$ (or $C^*_r(G) \cong C^*_r(H)$) as ...
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53 views

Difference between Schmidt decomposition and singular value decomposition

Schmidt decomposition of an operator is a useful tool of quantum information theory nowadays. Let $O$ be an operator acting on the Hilbert space $\mathcal{H}_{d_1} \otimes \mathcal{H}_{d_1}$. ...